HomeHome Metamath Proof Explorer
Theorem List (p. 258 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlnoppnhpg 25701* If two points lie on the opposite side of a line 𝐷, they are not on the same half-plane. Theorem 9.9 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵)
 
Theoremhpgerlem 25702* Lemma for the proof that the half-plane relation is an equivalence relation. Lemma 9.10 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑 → ∃𝑐𝑃 𝐴𝑂𝑐)
 
Theoremhpgid 25703* The half-plane relation is reflexive. Theorem 9.11 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐴)
 
Theoremhpgcom 25704* The half-plane relation commutes. Theorem 9.12 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
 
Theoremhpgtr 25705* The half-plane relation is transitive. Theorem 9.13 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)    &   (𝜑𝐶𝑃)    &   (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐶)       (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐶)
 
Theoremcolopp 25706* Opposite sides of a line for colinear points. Theorem 9.18 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐴𝑂𝐵 ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∧ ¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷)))
 
Theoremcolhp 25707* Half-plane relation for colinear points. Theorem 9.19 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   𝐾 = (hlG‘𝐺)       (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾𝐶)𝐵 ∧ ¬ 𝐴𝐷)))
 
Theoremhphl 25708* If two points are on the same half-line with endpoint on a line, they are on the same half-plane defined by this line. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐵𝐷)    &   (𝜑𝐵(𝐾𝐴)𝐶)       (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐶)
 
15.2.15  Midpoints and Line Mirroring
 
Syntaxcmid 25709 Declare the constant for the midpoint operation.
class midG
 
Syntaxclmi 25710 Declare the constant for the line mirroring function.
class lInvG
 
Definitiondf-mid 25711* Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 25715, midbtwn 25716, and midcgr 25717. (Contributed by Thierry Arnoux, 9-Jun-2019.)
midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
 
Definitiondf-lmi 25712* Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 25724. (Contributed by Thierry Arnoux, 1-Dec-2019.)
lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
 
Theoremmidf 25713 Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃)
 
Theoremmidcl 25714 Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃)
 
Theoremismidb 25715 Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝑀𝑃)       (𝜑 → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀))
 
Theoremmidbtwn 25716 Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ (𝐴𝐼𝐵))
 
Theoremmidcgr 25717 Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶)       (𝜑 → (𝐶 𝐴) = (𝐶 𝐵))
 
Theoremmidid 25718 Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)
 
Theoremmidcom 25719 Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))
 
Theoremmirmid 25720 Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝑀𝑃)       (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵)))
 
Theoremlmieu 25721* Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
 
Theoremlmif 25722 Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀:𝑃𝑃)
 
Theoremlmicl 25723 Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑀𝐴) ∈ 𝑃)
 
Theoremislmib 25724 Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐵 = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
 
Theoremlmicom 25725 The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐴) = 𝐵)       (𝜑 → (𝑀𝐵) = 𝐴)
 
Theoremlmilmi 25726 Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑀‘(𝑀𝐴)) = 𝐴)
 
Theoremlmireu 25727* Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ∃!𝑏𝑃 (𝑀𝑏) = 𝐴)
 
Theoremlmieq 25728 Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐴) = (𝑀𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremlmiinv 25729 The invariants of the line mirroring lie on the mirror line. Theorem 10.8 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))
 
Theoremlmicinv 25730 The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐴𝐷)       (𝜑 → (𝑀𝐴) = 𝐴)
 
Theoremlmimid 25731 If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝐵)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑀𝐶) = (𝑆𝐶))
 
Theoremlmif1o 25732 The line mirroring function 𝑀 is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀:𝑃1-1-onto𝑃)
 
Theoremlmiisolem 25733 Lemma for lmiiso 25734. (Contributed by Thierry Arnoux, 14-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝑍)    &   𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵)))       (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
 
Theoremlmiiso 25734 The line mirroring function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 10.10 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
 
Theoremlmimot 25735 Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀 ∈ (𝐺Ismt𝐺))
 
Theoremhypcgrlem1 25736 Lemma for hypcgr 25738, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐵 = 𝐸)    &   𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))    &   (𝜑𝐶 = 𝐹)       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremhypcgrlem2 25737 Lemma for hypcgr 25738, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐵 = 𝐸)    &   𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremhypcgr 25738 If the catheti of two right-angled triangles are congruent, so is their hypothenuse. Theorem 10.12 of [Schwabhauser] p. 91. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremlmiopp 25739* Line mirroring produces points on the opposite side of the mirroring line. Theorem 10.14 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐷 ∈ ran 𝐿)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   (𝜑𝐴𝑃)    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑𝐴𝑂(𝑀𝐴))
 
Theoremlnperpex 25740* Existence of a perpendicular to a line 𝐿 at a given point 𝐴. Theorem 10.15 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐷 ∈ ran 𝐿)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝐷)    &   (𝜑𝑄𝑃)    &   (𝜑 → ¬ 𝑄𝐷)       (𝜑 → ∃𝑝𝑃 (𝐷(⟂G‘𝐺)(𝑝𝐿𝐴) ∧ 𝑝((hpG‘𝐺)‘𝐷)𝑄))
 
Theoremtrgcopy 25741* Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: existence part. First part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 4-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
 
Theoremtrgcopyeulem 25742* Lemma for trgcopyeu 25743. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)       (𝜑𝑋 = 𝑌)
 
Theoremtrgcopyeu 25743* Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: uniqueness part. Second part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → ∃!𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
 
15.2.16  Congruence of angles
 
Syntaxccgra 25744 Declare the constant for the congruence between angles relation.
class cgrA
 
Definitiondf-cgra 25745* Define the congruence relation bewteen angles. As for triangles we use "words of points". See iscgra 25746 for a more human readable version. (Contributed by Thierry Arnoux, 30-Jul-2020.)
cgrA = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝𝑚 (0..^3)) ∧ 𝑏 ∈ (𝑝𝑚 (0..^3))) ∧ ∃𝑥𝑝𝑦𝑝 (𝑎(cgrG‘𝑔)⟨“𝑥(𝑏‘1)𝑦”⟩ ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))})
 
Theoremiscgra 25746* Property for two angles ABC and DEF to be congruent. This is a modified version of the definition 11.3 of [Schwabhauser] p. 95. where the number of constructed points has been reduced to two. We chose this version rather than the textbook version because it is shorter and therefore simpler to work with. Theorem dfcgra2 25766 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)))
 
Theoremiscgra1 25747* A special version of iscgra 25746 where one distance is known to be equal. In this case, angle congruence can be written with only one quantifier. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &    = (dist‘𝐺)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
 
Theoremiscgrad 25748 Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑋𝐸𝑌”⟩)    &   (𝜑𝑋(𝐾𝐸)𝐷)    &   (𝜑𝑌(𝐾𝐸)𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
Theoremcgrane1 25749 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐴𝐵)
 
Theoremcgrane2 25750 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐵𝐶)
 
Theoremcgrane3 25751 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐸𝐷)
 
Theoremcgrane4 25752 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐸𝐹)
 
Theoremcgrahl1 25753 Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of [Schwabhauser] p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &   (𝜑𝑋(𝐾𝐸)𝐷)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑋𝐸𝐹”⟩)
 
Theoremcgrahl2 25754 Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of [Schwabhauser] p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &   (𝜑𝑋(𝐾𝐸)𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
 
Theoremcgracgr 25755 First direction of proposition 11.4 of [Schwabhauser] p. 95. Again, this is "half" of the proposition, i.e. only two additional points are used, while Schwabhauser has four. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋(𝐾𝐵)𝐴)    &   (𝜑𝑌(𝐾𝐵)𝐶)    &   (𝜑 → (𝐵 𝑋) = (𝐸 𝐷))    &   (𝜑 → (𝐵 𝑌) = (𝐸 𝐹))       (𝜑 → (𝑋 𝑌) = (𝐷 𝐹))
 
Theoremcgraid 25756 Angle congruence is reflexive. Theorem 11.6 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
 
Theoremcgraswap 25757 Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐶𝐵𝐴”⟩)
 
Theoremcgrcgra 25758 Triangle congruence implies angle congruence. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
Theoremcgracom 25759 Angle congruence commutes. Theorem 11.7 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
 
Theoremcgratr 25760 Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐻𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝐽𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
 
Theoremcgraswaplr 25761 Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐹𝐸𝐷”⟩)
 
Theoremcgrabtwn 25762 Angle congruence preserves flat angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))       (𝜑𝐸 ∈ (𝐷𝐼𝐹))
 
Theoremcgrahl 25763 Angle congruence preserves null angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴(𝐾𝐵)𝐶)       (𝜑𝐷(𝐾𝐸)𝐹)
 
Theoremcgracol 25764 Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
 
Theoremcgrancol 25765 Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
 
Theoremdfcgra2 25766* This is the full statement of definition 11.2 of [Schwabhauser] p. 95. This proof serves to confirm that the definition we have chosen, df-cgra 25745 is indeed equivalent with the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴𝐵𝐶𝐵) ∧ (𝐷𝐸𝐹𝐸) ∧ ∃𝑎𝑃𝑐𝑃𝑑𝑃𝑓𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 𝑎) = (𝐸 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 𝑐) = (𝐸 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 𝑑) = (𝐵 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 𝑓) = (𝐵 𝐶))) ∧ (𝑎 𝑐) = (𝑑 𝑓)))))
 
Theoremsacgr 25767 Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐵 ∈ (𝐴𝐼𝑋))    &   (𝜑𝐸 ∈ (𝐷𝐼𝑌))    &   (𝜑𝐵𝑋)    &   (𝜑𝐸𝑌)       (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
 
Theoremoacgr 25768 Vertical angle theorem. Vertical, or opposite angles are the facing pair of angles formed when two lines intersect. Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. We follow the same path. Theorem 11.14 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 27-Sep-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &   (𝜑𝐵 ∈ (𝐶𝐼𝐹))    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑𝐵𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐵𝐹”⟩)
 
Theoremacopy 25769* Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))       (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
 
Theoremacopyeu 25770 Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. Akin to a uniqueness theorem, this states that if two points 𝑋 and 𝑌 both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)       (𝜑𝑋(𝐾𝐸)𝑌)
 
15.2.17  Angle Comparisons
 
Syntaxcinag 25771 Extend class relation with the geometrical "point in angle" relation.
class inA
 
Syntaxcleag 25772 Extend class relation with the "angle less than" relation.
class
 
Definitiondf-inag 25773* Definition of the geometrical "in angle" relation. (Contributed by Thierry Arnoux, 15-Aug-2020.)
inA = (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑𝑚 (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
 
Theoremisinag 25774* Property for point 𝑋 to lie in the angle ⟨“𝐴𝐵𝐶”⟩ Defnition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺𝑉)       (𝜑 → (𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ((𝐴𝐵𝐶𝐵𝑋𝐵) ∧ ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵𝑥(𝐾𝐵)𝑋)))))
 
Theoreminagswap 25775 Swap the order of the half lines delimiting the angle. Theorem 11.24 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)       (𝜑𝑋(inA‘𝐺)⟨“𝐶𝐵𝐴”⟩)
 
Theoreminaghl 25776 The "point lie in angle" relation is independent of the points chosen on the half lines starting from 𝐵. Theorem 11.25 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 27-Sep-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    &   (𝜑𝐷𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐷(𝐾𝐵)𝐴)    &   (𝜑𝐹(𝐾𝐵)𝐶)    &   (𝜑𝑌(𝐾𝐵)𝑋)       (𝜑𝑌(inA‘𝐺)⟨“𝐷𝐵𝐹”⟩)
 
Definitiondf-leag 25777* Definition of the geometrical "angle less than" relation. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
= (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑𝑚 (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
 
Theoremisleag 25778* Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
𝑃 = (Base‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))
 
Theoremcgrg3col4 25779* Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.)
𝑃 = (Base‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))       (𝜑 → ∃𝑦𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
 
15.2.18  Congruence Theorems
 
Theoremtgsas1 25780 First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then third sides are equal in length. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))
 
Theoremtgsas 25781 First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
Theoremtgsas2 25782 First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐴𝐶)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
 
Theoremtgsas3 25783 First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐴𝐶)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)
 
Theoremtgasa1 25784 Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)       (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
 
Theoremtgasa 25785 Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
Theoremtgsss1 25786 Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
Theoremtgsss2 25787 Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
 
Theoremtgsss3 25788 Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)
 
Theoremisoas 25789 Congruence theorem for isocele triangles: if two angles of a triangle are congruent, then the corresponding sides also are. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐵”⟩)       (𝜑 → (𝐴 𝐵) = (𝐴 𝐶))
 
15.2.19  Equilateral triangles
 
Syntaxceqlg 25790 Declare the class of equilateral triangles.
class eqltrG
 
Definitiondf-eqlg 25791* Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.)
eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
 
Theoremiseqlg 25792 Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))
 
Theoremiseqlgd 25793 Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐵 𝐶))    &   (𝜑 → (𝐵 𝐶) = (𝐶 𝐴))    &   (𝜑 → (𝐶 𝐴) = (𝐴 𝐵))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺))
 
15.3  Properties of geometries
 
15.3.1  Isomorphisms between geometries
 
Theoremf1otrgds 25794* Convenient lemma for f1otrg 25796. (Contributed by Thierry Arnoux, 19-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌)))
 
Theoremf1otrgitv 25795* Convenient lemma for f1otrg 25796. (Contributed by Thierry Arnoux, 19-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
 
Theoremf1otrg 25796* A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝐻𝑉)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑 → (LineG‘𝐻) = (𝑥𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐽𝑦) ∨ 𝑥 ∈ (𝑧𝐽𝑦) ∨ 𝑦 ∈ (𝑥𝐽𝑧))}))       (𝜑𝐻 ∈ TarskiG)
 
Theoremf1otrge 25797* A bijection between bases which conserves distances and intervals conserves also the property of being a Euclidean geometry. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐷 = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐵 = (Base‘𝐻)    &   𝐸 = (dist‘𝐻)    &   𝐽 = (Itv‘𝐻)    &   (𝜑𝐹:𝐵1-1-onto𝑃)    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))    &   ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))    &   (𝜑𝐻𝑉)    &   (𝜑𝐺 ∈ TarskiGE)       (𝜑𝐻 ∈ TarskiGE)
 
15.4  Geometry in Hilbert spaces
 
Syntaxcttg 25798 Function to convert an algebraic structure to a Tarski geometry.
class toTG
 
Definitiondf-ttg 25799* Define a function converting a subcomplex Hilbert space to a Tarski Geometry. It does so by equipping the structure with a betweenness operation. Note that because the scalar product is applied over the interval (0[,]1), only spaces whose scalar field is a superset of that interval can be considered. (Contributed by Thierry Arnoux, 24-Mar-2019.)
toTG = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥))}) / 𝑖((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
 
Theoremttgval 25800* Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    &    = (-g𝐻)    &    · = ( ·𝑠𝐻)    &   𝐼 = (Itv‘𝐺)       (𝐻𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >